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arXiv:1305.3641 · Annales Henri Poincaré

Excitation Spectrum of Interacting Bosons in the Mean-Field Infinite-Volume Limit

Jan Dereziński, Marcin Napiórkowski

Published Jan 9, 2014• 67 citations
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Abstract

We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011 ) to large volumes.

DOI
10.1007/s00023-013-0302-4

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We consider homogeneous Bose gas in a large cubic box with periodic boundary conditions, at zero temperature. We analyze its excitation spectrum in a certain kind of a mean-field infinite-volume limit. We prove that under appropriate conditions the excitation spectrum has the form predicted by the Bogoliubov approximation. Our result can be viewed as an extension of the result of Seiringer (Commun. Math. Phys. 306:565–578, 2011 ) to large volumes.
In 1947, Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely, to test the validity of Bogoliubov’s predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes, we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.
We consider second-quantized homogeneous Bose gas in a large cubic box with periodic boundary conditions at zero temperature. We discuss the energy-momentum spectrum of the Bose gas and its physical significance. We review various rigorous and heuristic results as well as open conjectures about its properties. Our main aim is to convince the readers, including those with mainly mathematical background, that this subject has many interesting problems for rigorous research. In particular, we investigate the upper bound on the infimum of the energy for a fixed total momentum k given by the expectation value of one-particle excitations over a squeezed states. This bound can be viewed as a rigorous version of the famous Bogoliubov method. We show that this approach seems to lead to a (nonphysical) energy gap. The variational problem involving squeezed states can serve as the preparatory step in a perturbative approach that should be useful in computing excitation spectrum. This version of a perturbative approach to the Bose gas seems (at least in principle) superior to the commonly used approach based on the c-number substitution.
An extension to nonzero temperatures of the Hugenholtz-Pines procedure for the degenerate Bose gas is provided. It is shown that under certain conditions the elementary excitation spectrum must approach zero for zero momentum, at nonzero, as well as at zero temperatures. The apparent discrepancy of this result with the particular case of a charged boson gas with a uniform positive background is discussed. Finally, a suggestion is made for modifying the Bogoliubov approximation, which is particularly relevant at nonzero temperatures.
We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.
This thesis is devoted to a rigorous study of interacting quantum gases. The main objects of interest are the closely related concepts of excitation spectrum and quasiparticles. The immediate motivation of this work is to propose a spectral point of view concerning these two concepts. In the first part of this thesis we discuss the concepts of excitation spectrum and quasiparticles. We provide an overview of physical motivations and based on that we propose a spectral and Hamiltonian-based approach towards these terms. Based on that, we formulate definitions and propositions related to these concepts. In the second part we recall the Bogoliubov and Hartree-Fock-Bogoliubov approximations, which in the physics literature are used to obtain the quasiparticle picture. We show how these two approaches fit into a universal scheme which allows us to arrive at a quasiparticle picture in a more general setup. This scheme is based on the minimization of Hamiltonians over the so-called Gaussian states. Its abstract formulation is the content of Beliaev's Theorem. In the last part we present a rigorous result concerning the justification of the Bogoliubov approximation. This justification employs the concept of the mean-field and infinite-volume limit. We show that for a large number of particles, a large volume and a sufficiently high density, the low-lying energy-momentum spectrum of the homogeneous Bose gas is well described by the Bogoliubov approximation. This result, which is formulated in the form of a theorem, can be seen as the main result of this thesis.